Every few years it's worth tweaking an idea. In Pre-Cal there's a section on the three dimensional coordinate system, and three dimensional vectors. Incredibly hard to visualize with a two dimensional surface. Some years ago my teaching neighbor cooked up a coordinate system with a bunch of meter sticks rubber banded together. Worked out ok.

A couple of years later I bought some yarn and made a room sized version. Later on I had the kids actually do something inside of the space and had them wander around placing coordinates in appropriate locations. Then some days later, they built their own coordinate system. It started in large scale, evolved, and reached a natural conclusion:

The students did a fine job with the work. They built the cubes in pairs and made an effort to understand how the octants oriented with each other. But, they were all producing the same thing, and merely following a blueprint I had provided.

This year, I removed the script. Given 40 straws (5" long), electrical tape, and a partner:

- build a three dimensional object
- construct an xyz reference marker using the right hand rule as a guide
- place the marker inside your object at any location
- flag 10 coordinates with the appropriate sign convention based on your reference
- mount the object to the lockers

The open ended nature of the task meant allowing more time. Previously, the cubes could be completed in 90 minutes (10 minutes of instruction, 80 minutes of work). This year, it took about 15 minutes for setup and instruction, and the students spent 75 minutes building. The following class period (50 minutes long), they tightened up their structures and marking the coordinates.

I've learned that for time consuming projects like this, instruction is best given in small chunks. The students were going to spend a long time building and would quickly forget any long winded explanation about coordinate labeling. Building became Day 1 and Labeling became Day 2.

The creativity on display was amazing. Geometric solids, Minecraft people, animals, rockets, houses, and a wrecking ball were among the myriad of entries. The labeling added the thinking element missing from reproducing a basic coordinate system. Students had to consider whether negative z even existed inside their object based on their reference location. Many students had to contend with 0 value in particular dimensions. Which value is 0? Why would that be? Setting an arbitrary location for their reference (whether they were consciously or not) helped with understanding the relative nature of coordinate systems. If I place it here, my object has entirely positive dimensions, move it this way and now I have a region with negative y.

Bravo to the students. Some day in the future this results in a bunch of 3D printed objects but given my current resources, this worked well.